� � � ��d k θ ↦ → θ1,...,θk, θj − fj {θi} n

3.9 A likelihood ratio test for nested composite hypotheses: Wilks's theorem. Let Θ be a d-dimensional parameter space, specifically, an open set in R d. Let H 0 be a k-dimensional subset of Θ, in a sense to be made more precise below, for some k < d. For example, H 0 could be the intersection with Θ of a k-dimensional flat hyperplane. Let {P θ , θ ∈ Θ} be an equivalent family of laws on a sample space (X, B) with a likelihood function f (θ, x) > 0 for all θ ∈ Θ and x ∈ X. Assume that observations X 1 ,. .. , X n are i.i.d. P θ for some θ ∈ Θ. We want to test the hypothesis that θ ∈ H 0. S. S. Wilks proposed the following test: let L(θ, x) := log f (θ, x) be the log likelihood. For n observations, let the maximum log likelihoods over Θ and H 0 be respectively n n MLL d := sup L(θ, X j), MLL k := sup L(θ, X j). θ∈Θ j=1 θ∈H 0 j=1 Let W := 2(MLL d − MLL k). Wilks found that if the hypothesis H 0 is true, then the distribution of W converges as n → ∞ to a χ 2 distribution with d − k degrees of freedom, not depending on the true θ = θ 0 ∈ H 0. Thus, H 0 would be rejected if W is too large in terms of the tabulated χ 2 distribution. d−k It turns out that Wilks's conclusion can be proved under the same assumptions as are used to prove the lower bounds on asymptotic efficiency of estimators in Section 3.7 and efficiency of maximum likelihood estimators in Section 3.8. It will be said that H 0 is a k-dimensional C 2 imbedded submanifold of Θ for some k < d if for each θ ∈ H 0 , after a translation of coordinates taking θ to 0 and a suitable rotation of coordinates, H 0 has a tangent hyperplane K 0 at 0 given by θ k+1 = · · · = θ d = 0, meaning that the intersection k of H 0 with a neighborhood V of 0 is given by θ j = f j ({θ i } i=1) for j = k + 1,. .. , d, where f j …